Download On the complexity of case-based planning

Journal of Experimental and Theoretical Artificial Intelligence
Vol. 00, No. 00, July 2004, 1–16
On the complexity of case-based planning
P. LIBERATORE
Università di Roma “La Sapienza”, Italy
(Received: June 2005)
We analyze the computational complexity of problems related to case-based planning: planning
when a plan for a similar instance is known, and planning from a library of plans. We prove that
planning from a single case has the same complexity than generative planning (i.e. planning ‘from
scratch’); using an extended definition of cases, complexity is reduced if the domain stored in the case
is similar to the one to search plans for. Planning from a library of cases is shown to have the same
complexity. In both cases, the complexity of planning remains, in the worst case, PSPACE-complete.
Keywords: automated planning, case-based planning, computational complexity.
1
Introduction
Case-based reasoning (Kolodner, 1993,Aamodt and Plaza, 1994,Watson, 1997)
is a problem solving methodology based on using a library of solutions for similar problems, i.e. a library of ‘cases’ with their respective solutions. Roughly
speaking, case-based planning consists into storing generated plans and using
them for finding new plans (Hammond, 1989, Bergmann et al., 1998, Spalazzi,
2001). In practice, what is stored is not only a specific problem with a specific
solution, but also some additional information that is considered useful to the
aim of solving new problems, e.g. information about how the plan has been
derived (Veloso, 1994), why it works (Kambhampati, 1990b, Hanks and Weld,
1995), when it would not work (Ihrig and Kambhampati, 1997), etc. Different
case-based planners differ on how they store cases, which cases they retrieve
when the solution of a new problem is needed, how they adapt a solution to a
new problem, whether they use one or more cases for building a new solution,
etc. The survey papers by Bergmann et al. (1998) and by Spalazzi (2001) give
a detailed introduction to case-based planning.
In this paper, we study the computational complexity of case-based planning, i.e. we characterize the complexity of case-based planning using the
theory of the NP-completeness (Johnson, 1990). What makes this analysis
different from the other results on the complexity of planning (Bylander,
1991, Nebel et al., 1997, Nebel and Bäckström, 1994, Bäckström and Nebel,
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1995, Bäckström, 1992, Bäckström and Nebel, 1992, Bäckström and Nebel,
1993, Bäckström and Jonsson, 1995). is that case-based planning is not actually a problem, but rather a family of solutions for a problem. In fact, the
theory of computational complexity can only characterize the complexity of
problems, not of family of solutions. On the other hand, case-based planning
always requires solving some specific subproblems, such as, for example, the
adaptation of a plan to a different problem. In this paper, we study the complexity of some problems that are related to case-based planning.
The first problem we consider is that of finding a plan given another plan
that works on a slightly different domain. This problem formalizes the task of
plan adaptation that case-based planners have to perform. While the formalization of this problem disregards many aspect of plan adaptation in practice
(e.g. more plans for similar cases may be available), it nevertheless contains the
main requirement of plan adaption: building a new plan given an old one. This
problem has been already analyzed by Nebel and Koehler (1993, 1995) under
the constraint of minimal plan change: the new plan should be as similar to the
old one as possible; plan adaptation in this case is called conservative. There
are some scenarios in which this constraint is reasonable (e.g. we schedule the
actions of the old plan before starting planning, and not executing them results
in an additional cost); while conservatism is sometimes obtained as a byproduct of algorithms that work by changing an old plan (Kambhampati, 1990b),
it is usually not a requirement. On the contrary, not enforcing minimal change
has been considered a viable alternative for escaping the high complexity of
conservative plan adaptation (Au et al., 2002). The absence of the constraint
of conservatism is also evident in the generative case-based planning approach,
which differs from the transformational case-based approach outlined above in
that the new plan is generated from scratch (rather than from the old plan),
and the case is used to guide the choices made during the search (Bergmann
et al., 1998).
The main problem of analyzing a solution technique, rather than a specific
problem, is that the implemented solutions may greatly differ to each other.
While the abstract idea of reusing old plans is part of all case-based planners,
even giving a common definition of ‘an old plan’ is not easy. This may be
simply a specific plan, but can also be an abstract plan, or a partially ordered
plan, and may include additional information (Kambhampati, 1990b, Veloso,
1994, Ihrig and Kambhampati, 1997). Moreover, the plan adaptation problem
may not be tackled at all: planning by derivational analogy (Carbonell, 1986,
Veloso and Carbonell, 1993) uses the traces of the search done for finding a
plan, rather than the plan itself; generative case-based planners (Bergmann et
al., 1998) build a plan from scratch by using information from more than one
case, rather than adapting a specific plan. In the opinion of the author of the
present paper, the problem of plan adaptation in its simpler formalization is
On the complexity of case-based planning
3
a good starting point for the computational analysis of plan adaptation, even
if it is not the problem that is faced in practice.
What makes giving a formalization that is at the same time general and
precise a difficult task is that plan adaptation cannot always be separated
from the other steps of case-based planning. For example, the plan to adapt is
not chosen arbitrarily in the library of plans; it is chosen because it is expected
to be adaptable to the new situation; moreover, more cases may be selected,
leading to more than one ‘starting point’ for the search of the new plan. Being
impossible to precisely formalize the whole case-based planning process and
remain general enough, we consider the problem of case-based planning in a
very general way, in which the only assumption that is made is that a case, or
a library of cases, is a data structure of polynomial size.
The problems that are therefore considered are: planning from a specific
known plan, and planning from a ‘generalized’ case or a library of cases. The
first problem formalizes the basic plan adaptation step; the second problem
formalizes the fact that cases usually contain additional information other than
the plan itself; the third problem is a formalization of the whole case-based
planning process.
In the analysis of these problems, we do not enforce the constraint of minimal change. The resulting freedom in the search for plans may simplify the
algorithms (Au et al., 2002), but makes the computational analysis more complicated. Indeed, the theory of NP-completeness formalizes problems that can
be expressed in a scheme like the following one:
Instance: a planning domain;
Question: is there a plan for the domain?
(for some reason not clear to the author, the words ‘instance’ and ‘question’ are
often written in small caps font.) The fact that this is a decision problem (the
solution can only be yes or not) is not a big restriction, as search problems
(where the solution can be a more complex data structure) can be usually
reduced to a number of decision problems. What makes this formalization
restrictive in the setting of plan adaptation is that there is no slot in the
scheme where to place the old plan. Indeed, the old plan is not exactly part
of the instance, as it presence does not change the definition of the problem.
As well, it is not part of the question. The plan adaptation problem would be
better formalized by a list of three parts:
Instance: a planning domain;
Hint: a plan for a similar domain;
Question: is there a plan for the new domain?
This is still a decision problem (its solution is either yes or not). However, its
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definition contains a new part, the hint, which is not necessary for answering
the question, but can be useful nevertheless. In other words, the existence of
a plan only depends on the domain, not on the hint. The difference between
the hint and the instance makes the analysis of plan adaptation carried on
in this paper different from that of conservative plan adaptation: in the latter
problem, the plan for the similar domain is part of the instance, as the new plan
to be found depends on the old one (Nebel and Koehler, 1993). In other words,
the instance of conservative plan adaptation contains a plan for a domain, and
the question is whether there exists a similar plan for the new domain. Roughly
speaking, an hint is something we could disregard while looking for a solution,
while the old plan cannot in the case of conservative plan adaptation.
The problems we analyze in this paper are the problem of adaptation from a
specific plan, from a plan plus other information, and the problem of planning
with a library of cases. In all these cases, we are given an instance of the planning problem and an hint, which can be a plan, a plan plus other information,
or a plan library. The technical results of this paper is that plan adaptation
of a single specific plan can be as hard as generative planning (i.e. planning
‘from scratch’ disregarding the hint completely); planning from a more general definition of cases may be easier than generative planning in some cases,
but is as hard in general. The same results hold for case-based planning, i.e.
planning from a library of cases.
2
Preliminaries
The planning problems analyzed in this paper are formalized using the propositional STRIPS formalism. A STRIPS instance (or a STRIPS domain) is a
4-tuple hP, O, I, Gi, where P is the set of conditions, O is the set of operators,
I is the initial state, and G is the goal. The conditions are facts that can be
true or false in the world of interest. A state S is a set of conditions, and
represents the state of the world in a certain time point. The conditions in S
are those representing facts that are true in the world, while those not in S
represent facts currently false.
The initial state is a state, thus a set of conditions. The goal is specified by
giving a set of conditions that should be achieved, and another set specifying
which conditions should not be made true. Thus, a goal G is a pair hM, N i,
where M is the set of conditions that should be made true and N is the set
of conditions that should be made false.
The operators are actions that can be performed to achieve the goal. Each
operator is a 4-tuple hφ, η, α, βi, where φ, η , α, and β are sets of conditions.
When executed, such an operator makes the conditions in α true, and those
in β false, but only if the conditions in φ are currently true and those in η are
On the complexity of case-based planning
5
currently false. The conditions in φ and η are called the positive and negative
preconditions of the operator. The conditions in α and β are called the positive
and negative effects or postconditions of the operator.
Given an instance of a STRIPS planning domain hP, O, I, Gi, we define a
plan for it as a sequence of operators that, when executed in sequence from
the initial state, lead to a state where all the conditions in M are true and
all those in N are false. More details about the definition of STRIPS can be
found in other papers (Fikes and Nilsson, 1971, Bylander, 1991).
A planning case is a pair hDo , Po i, where Do is a planning instance and
Po is a plan for it. Plan adaptation is formalized as follows: we are given
a plan case and a specific domain D that contains the same conditions and
actions of the domain Do . The problem is that of finding a plan for D. This
is a simplification of the plan adaptation step of case-based planners, but
contains all main components: the case with a solution and a new instance
to be solved. Somehow, we are studying this problem in isolation from the
general case-based planning process, as for example we disregard the fact that
the case is typically chosen in such a way P0 is expected to be useful for
finding a plan for D. We then consider the problem of plan adaptation in the
assumption that the case is not only a pair hDo , Po i but may also contain other
information; we abstract over the kind of information that is stored, so that our
results are independent on whether we use derivational traces, abstractions,
previous failures, etc. We prove that planning adaptation is feasible, in this
case, provided that the domain of the case and the current domain are similar.
The second problem we consider is whether a library of plans can improve
the efficiency of finding a plan for a new domain. A simple (and somehow
simplistic) definition of a plan library is a set of cases {hDi , Pi i | 1 ≤ i ≤ m}.
We consider this definition of plan library as a special case of some importance,
but the result we prove holds for any kind of plan library. This generalization is
necessary because a case library usually contains much more information than
just a set of planning instances with their relative solutions (e.g. plans may be
abstract, they may be partially ordered, the library may contains indexes using
for example description logics (Koehler, 1996), the cases themselves may be
abstract and stored hierarchically, etc.) By assuming that the library of cases
is an arbitrary data structure, our results carry on to all specific classes of plan
libraries. We remark that the complexity-theoretic results for this generalized
definition of case library are more related to the generative case-based planning
than to the transformational approach, in that the plan library is only assumed
to be used when it is useful in the process of searching for a new plan, and is
not necessarily used to provide a starting point of the search process.
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Results
The problem of deciding whether there exists a plan for a STRIPS instance
is denoted as PLANSAT, and is known to be PSPACE-complete (Bylander,
1991, Bylander, 1994). If a planning case hDo , Po i is also known, the problem
cannot become more difficult, as we can simply disregard the case and find
the plan using D only. Essential to this trivial result is that we do not enforce
adaptation to be conservative, as motivated in the Introduction. The problem
of plan adaptation is therefore in PSPACE. While a problem cannot be made
harder by the presence of an hint, it may become easier. The following theorem
shows that this is not the case for plan adaptation.
Theorem 3.1 Deciding whether there exists a plan for a STRIPS instance
D, given a case hDo , Po i, is PSPACE-complete, even if Do and D only differ
from one condition of the initial state.
Proof. The problem is in PSPACE, as we can find the a plan for D disregarding
the case hDo , Po i.
We prove that the problem of plan adaptation is PSPACE-hard by showing
a reduction from PLANSAT. In other words, we show that, given an instance
of PLANSAT hP, O, I, Gi, there exists hDo , Po i and D such that Po is a plan
for Do , the domains Do and D only differ for one condition of the initial state,
and D has a plan if and only if hP, O, I, Gi has a plan.
The domains Do and D are obtained from hP, O, I, Gi by adding a single
condition a, a single action e = h{a}, ∅, M, N i, where G = hM, N i is the goal
of the original instance, and possibly modifying the initial state:
Do = hP ∪ {a}, O ∪ {e}, I ∪ {a}, Gi
D = hP ∪ {a}, O ∪ {e}, I, Gi
The plan Po that is part of the case is hei. It can be easily verified that Po
is a plan for Do (its preconditions are verified in the initial state of Do , and
its consequences are exactly the goal.) As a result, hDo , Po i and D constitute
a valid plan adaptation instance.
A case-based planner, while looking for a plan for D, would first check
whether Po works in D and then try to adapt it if it does not. In this case,
Po is not executable at all in D, as the precondition a of e is not true in the
initial state of D. Moreover, the plan Po is completely useless for solving D, as
it requires a to be true, while a is not not initially true in D and cannot even
be made true because no action in O ∪ {e} makes it true. As a result, a can be
removed altogether from D, along with e that is never executable. After this
change, D becomes hP, O, I, Gi, which is exactly the original instance. We can
therefore conclude that D has a plan if and only if the original instance has a
On the complexity of case-based planning
7
plan, thus proving the PSPACE-hardness of the plan adaptation problem.
This theorem proves that checking the existence of a case hDo , Po i does
not simplify the check of existence of a plan for a domain D that is similar
to Do . Namely, the problem remains PSPACE-complete. We remark that the
hardness part of this theorem holds even if the instances are similar, not only
in this case. This result is therefore relevant even in those settings in which
the choice of the case to reuse is not driven by the similarity of the domain to
the one to be solved (Kambhampati, 1990a).
Analyzing the proof, we observe that the reduction used for proving the
hardness part can be described as: ‘given a planning instance hP, O, I, Gi,
there exists a case hDo , Po i and a domain D such that...’. The important part
is that we have chosen the specific plan Po that is part of the case. In a way,
this theorem proves that there exists a plan for Do that does not simplify the
plan existence problem for D, i.e. some plans do not help.
A natural question is now: given that some plans do not help, may it be
that some other plans do? In other words, it may be that a sensible choice of
a plan Po for Do is useful for solving the problem for D. We give a negative
answer to this question.
Theorem 3.2 The problem of checking the existence of a plan for D, given
a case hDo , Po i, where Po is any plan of Do , is PSPACE-complete even if D
and Do only differ from one condition of the initial state.
Proof. The proof is based on the idea of reducing the PLANSAT problem to
the plan adaptation problem in which Po is the only plan of Do . By proving
that Po is not useful for solving PLANSAT for D, we prove that no plan is, in
general, useful.
The proof of Theorem 3.1 has the only problem that Po = hei may not be
the only plan of Do . We therefore modify the domain Do to make it so. Given
a PLANSAT instance hP, O, I, Gi, the domains D and Do are still based on
the set of conditions P ∪ {a}, where a is a new condition. The operators are
O0 ∪ {e}, where:
O0 = {hα, β ∪ {a}, γ, δi | hα, β, γ, δi ∈ O}
e = h{a}, ∅, M, N i
In words, we add a as a negative precondition of each operator in O. The
new action e is defined as in the proof of the previous theorem. The domains
D and Do are defined as:
Do = hP ∪ {a}, O0 , I ∪ {a}, Gi
D = hP ∪ {a}, O0 , I, Gi
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The plan Po is hei. This time, Po is the only plan for Do , as all other actions
have a as a negative preconditions, while a is initially true and no action makes
it false. On the other hand, the action e is not executable in D, as a is initially
false and no action makes it true. As a result, we can remove a and e from
D, and this makes it identical to the original instance. As a result, D as a
plan if and only if hP, O, I, Gi has a plan. This proves that the problem of
plan adaptation remains PSPACE-hard regardless of how the plan in the case
hDo , Po i is chosen.
This theorem proves that no plan for Do can be useful in finding a plan
for D, even if D and Do only differ for a single condition of the initial state.
This result, however, only holds when we assume that a case is exactly a
pair hDo , Po i, i.e. a domain and a plan for it. As explained in the Introduction, case-based planners usually record with plans other information such as
derivational traces, abstractions of the plan, information that anticipates when
the plan may fail, etc. This additional information can simplify the plan adaptation process. Since case-based planners greatly differ on what information is
recorded, we simply assume that the case is recorded as a pair hDo , Ao i, where
Ao is a polynomial-size data structure. We can prove that such cases may be
of help if the new domain to be solved is similar to the one in a case.
The theorem showing this fact, as well as the following one, requires a digression into the topic of problem preprocessing. Problems such as plan adaptation
can be formalized as set of pairs hDo , Di, where the question is: given Do , is
there any Ao such that solving the question of plan existence of D can be done
in polynomial time given hDo , Ao i? In terms of problem preprocessing, we are
given hDo , Di and ask whether preprocessing Do can result in a data structure
Ao that makes the PLANSAT problem on D easy to solve, i.e. polynomial-time.
Problems for which preprocessing lowers the complexity to P form the class
;P, a.k.a. comp-P, a.k.a. compilable to P. The following lemma shows that
the problem of plan adaptation is compilable to P.
Lemma 3.3 The problem of checking the plan existence of D is in ;P, given
hDo , Di, where Do is a domain that only differs from D for a constant number
of conditions in the initial state and goal.
Proof. Cadoli et al. (2002) proved that all problems for which a polynomial
number of varying parts (in this case, the number of possible D) that correspond to the same fixed part (in this case, Do ) is a constant, then the problem
is compilable to P. This is actually the case, as D and Do only differ for a
constant number c of conditions in the initial states or goal; as a result, for any
Do there are only |P|c possible D. Since c is a constant, this function |P|c is
polynomial in the size of Do and D. As a result, the problem can be compiled
to P, i.e. it is in ;P.
On the complexity of case-based planning
9
The following is an easy corollary of the above lemma.
Theorem 3.4 For every domain Do there exists a data structure Ao such
that hDo , Ao i allows for solving the problem of plan existence of a domain D
that only differs from Do for a constant number of conditions in the initial
state or the goal in polynomial time.
Proof. By the above lemma, the fixed part Do of the problem hDo , Di can be
preprocessed in such a way the result of this phase is polynomial in size and
allows for solving the problem of plan existence for D in polynomial time, if D
and Do only differ for a constant number of conditions of the initial state and
goal. As a result, for any given Do there is a polynomial-size data structure
Ao such that hDo , Ao i makes polynomial the problem PLANSAT for every D
that only differ to Do only for a constant number of conditions of the initial
state or goal.
We remark that this result is theoretical, in that it abstracts over the possible
‘extended representations’ of cases. How this result apply to the various specific
representations of cases is an open question.
The question of whether an extended representation of cases, or a whole
library of cases, is of help in finding a plan for a new domain that can differ
from the ones that have already been analyzed for a non-constant number of
conditions can be given a negative answer by showing that the problem of
solving PLANSAT on hDo , Di is not compilable to P. We indeed prove that
this problem is k;PSPACE-complete. This is proved by means of the following
lemma.
Lemma 3.5 For every operator o over P, the STRIPS instance hP, O, I, Gi
has a plan if and only if the instance hP ∪ {y}, O ∪ {o}, I 0 , Gi has a plan,
where y is a new condition not in P, the operator o0 is o with the addition
of y as a positive precondition, and I 0 is I or I ∪ {y} depending on whether
o ∈ O or not.
Proof. Since y is a new condition not in P , and is only mentioned as a
precondition of o0 , no operator change its value. As a result, it is true if and
only if it is true in the initial state. Moreover, y is positive in the initial state
if and only if o ∈ O. As a result, o0 is equivalent to o if o ∈ O, and cannot be
executed if o 6∈ O. Since the addition of y and o0 and the possible removal of
o are the only changes from the first instance to the second, the property of
plan existence does not change.
This lemma looks like a trivial property, but in fact it says something interesting about the complexity of the PLANSAT problem: regardless of whether
o ∈ O, we can let o0 to be in O, and use a variable y to encode whether o in O
or not. In other words, we are making an element of the set of operators fixed,
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as o0 is added to the set of operators regardless of whether o ∈ O. Since the
original problem can be reduced to this new one in which the set of operators
has a fixed part, the complexity of the new problem is at least as high as that
of the original problem. By iterating this procedure, we can make the set of
operators completely fixed. As a result, the complexity of the problem remains
the same even if the set of operators is fixed.
The formal proof uses a sufficient condition for proving that a problem is
non compilable to P called representative equivalence.
Lemma 3.6 The problem of determining a plan for D, given hDo , Di,
where Do and D contain the same conditions and operators, is k;PSPACEcomplete.
Proof. Membership follows from the fact that PLANSAT is in PSPACE, and
that any problem in PSPACE is also in k;PSPACE.
Hardness: we show a reduction from the PLANSAT problem when operators
are restricted to have two preconditions and two postconditions. The problem
of plan existence has been proved PSPACE-hard even under this restriction
(Bylander, 1994).
In order for using the condition of representative equivalence (Liberatore,
2001), we need first to show three functions for the PLANSAT problem: a
classification, representative, and extension function. We make the following
choice:
Class(hP, O, I, Gi) = |P|
Repr(n) = h{x1 , . . . , xn }, ∅, ∅, h∅, ∅ii
Exte(hP, O, I, Gi, m) = hP ∪ {x|P|+1 , . . . , xm }, O, I, Gi
In words, the class of an instance is the number of its conditions. The representative of the class n is the instance that has n conditions, no operators,
and empty initial state and goal (this instance has the empty plan h i as the
initial state satisfies the goal, but this is irrelevant.) Extending an instance
is obtained by simply adding conditions that are not then contained in any
operator nor in the initial state or goal. Technically, these new conditions are
false in the initial state and not required to have any specific value in the goal.
As a result, hP, O, I, Gi has a plan if and only if Exte(hP, O, I, Gi, m) has a
plan, as required for the extension function.
We now show a reduction that satisfies the condition of representative equivalence. By Lemma 3.5, we can replace O with a set of operators O0 which
contains a fixed operator o0 . By iterating this reduction for each possible operator of four conditions over P , we end up with an instance hP ∪ Y, OP , I 00 , Gi,
where OP is obtained from the set of all possible operators of four conditions
On the complexity of case-based planning
11
over P , and Y is a set of new conditions in correspondence with the operators
of OP .
Here we exploit the restriction on the number of preconditions and postconditions. The number of possible pairs of preconditions is given by n(n − 1);
since each precondition can be either positive or negative, we have exactly four
combinations. As a result, the number of possible preconditions of an operator
are 4n(n − 1), assuming that no operator contains the same precondition both
positively and negatively. For the same reason, there are exactly 4n(n − 1)
postconditions. As a result, there are exactly 16n2 (n − 1)2 possible operators
over n conditions. Therefore, the size of hP ∪ Y, OP , I 00 , Gi is polynomially
larger than the size of hP, O, I, Gi.
The instances that result from implementing this reduction from two instances that have the same conditions are the same. By renaming all conditions
to {x1 , . . . , xn }, the results of the reduction is the same if the two instances
have the same number of conditions. As a result, this reduction satisfies the
condition of representative equivalence, which is sufficient to show that the
problem that is reduced to (PLANSAT) is C-hard for any class C for which the
problem that is reduced from (PLANSAT again). As a result, the PLANSAT
problem is k;PSPACE-hard.
This proof, based on Lemma 3.5, has an intuitive construction: we can progressively make O fixed while maintaining the property of existence of plans.
As a result, the PLANSAT problem can be reduced to the PLANSAT problem where O is fixed, i.e. when O can be preprocessed. Since making the set
of conditions P fixed is only a matter of condition renaming, this reduction
shows that the complexity of the problem does not change even if P and O are
fixed, i.e. the fixed part of the instance is actually fixed. This method can be
used also for other problems. However, giving a general formulation is made
difficult by the fact that ‘making a part fixed’ depend on the specific problem
under analysis.
The result of k;PSPACE-hardness of PLANSAT implies that the knowledge
of a case does not reduce the complexity of plan existence, even if the case can
contain arbitrary data besides the plan.
Theorem 3.7 The problem of checking the plan existence for D does not
become polynomial even if a case hDo , Ao i is known, where Do has the same
conditions and operators of D, and Ao is an arbitrary polynomial-size data
structure depending only on Do , unless the polynomial hierarchy collapses.
Proof. Assume, on the contrary, that for every Do there exists a polynomialsized data structure Ao (i.e. the ‘extended plan’) that makes solving PLANSAT
on D a polynomial task. If this were the case, it would be possible to preprocess
Do obtaining Ao , as the preprocessing phase is not constrained in any way but
that its result must be of polynomial size. Since hDo , Ao i allows for solving
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PLANSAT on D in polynomial time, we have that the problem of plan existence
on hDo , Di is in ;P. Since the same problem is also k;PSPACE-hard, it
follows that k;PSPACE ⊆ ;P, which implies that PSPACE/poly ⊆ P/poly
thanks to a result by Cadoli et al. (2002), which in turns implies that Σp2 ∩ Πp2 =
PSPACE thanks to a result by Karp and Lipton (1980). In other words, the
polynomial hierarchy collapses to its second level.
This result can also be extended to the case in which a whole ‘library of
cases’ {hDi , Pi | 1 ≤ i ≤ m} is given, provided that all domains Di have the
same conditions and operators of the current domain D, and the whole library
is of polynomial size.
Theorem 3.8 The problem of checking the existence of plans for D cannot
be solved in polynomial time even if a library of plans {hDi , Pi | 1 ≤ i ≤ m} of
polynomial size is given, where all Di have the same conditions and operators
of D, unless the polynomial hierarchy collapses.
Proof. This is only a consequence of the above theorem: assume that
Ao = {hDi , Pi | 1 ≤ i ≤ m}; since hDo , Ao i does not make PLANSAT on
D polynomial-time, then {hDi , Pi | 1 ≤ i ≤ m} does not make it polynomialtime either.
As it is clear, some library of plans simplify the problem: for example, if the
library contains a case hDi , Pi i where D = Di , a plan for D is simply Pi . The
theorem indeed proves that such a simplification is not possible in general. As
it is also clear from the proof, using an extended definition of cases, in which
not only plans are recorded, is not useful either. This result requires some
discussion, which will be given in the next section.
4
Conclusions
In this paper, we have proved two kinds of results: first, adapting a specific plan
to a new domain is as hard as planning from scratch; second, a case or a library
of cases composed of domains, plans, and additional information, can simplify
the problem of planning in some cases, but remains hard in general. These
results are interesting because they are formally, and not only empirically,
proved.
The first kind of results are somehow not surprising, as it has already been
observed that even changing a single condition of the initial state or goal may
make a plan completely useless. Our results are simply formal proofs of this
observation.
The second kind of results are, in the author’s opinion, more interesting.
The fact that ‘extended’ cases sometimes lower the complexity of finding plans
On the complexity of case-based planning
13
formally validates the trend in case-based planning of storing complex information in cases, rather than simple plans. Nevertheless, this formal result only
holds in a very general settings, in which no assumption, besides polynomiality
of space, is made about what is stored in the case. How this result extends
to specific form of information used in case-based planners, such as abstract
plans or derivational traces, is an open question.
The negative results about the complexity of planning from an extended
plan or a library of plans are the most interesting ones of this paper. First,
they formally prove that case-based planning can be as hard as generative
planning; while this phenomena has been empirically observed (Bergmann et
al., 1998), it was not yet proved that it is intrinsic to the problem and not
related to the specific implementations. Second, compared with the result on
‘small changes’, it proves that there exists a trade-off of efficiency and size of
library in case-based planning. Again, what is interesting in this result is that
it has formally proved, not only empirically observed.
Let us compare the results of this paper with similar work in the literature.
The computational complexity of planning has been deeply investigated by
several authors (Bylander, 1991, Bylander, 1994, Nebel et al., 1997, Nebel and
Bäckström, 1994,Bäckström and Nebel, 1995,Bäckström, 1992,Bäckström and
Nebel, 1992, Bäckström and Nebel, 1993, Bäckström and Jonsson, 1995). All
these works are on planning from first principles, i.e. are about planning given
only a planning domain. More related to the present work are that by Nebel
and Koehler (1993, 1995), who analyzed the complexity of conservative plan
adaptation. Their complexity results are about planning when the constraint
to be similar to another known plan is enforced. While there are some scenarios where this constraint is important, it seems not to be enforced often
in case-based planning. Some work on plan adaptation has also been done by
Bylander (1996), who has shown that, probabilistically, plan modification is
simpler than plan generation provided that the domain of the case is similar
to the one to be solved.
As it has already been noticed in the introduction, the problem of plan
adaptation and that of planning with a library of cases is of a different kind
of most problems that are studied in computational complexity, in that the
data we are given include an ‘hint’ that can very well be neglected if necessary.
Such problem format include other problems, such as case-based reasoning in
general. The complexity of other forms of case-based reasoning is an interesting
problem which is however out of the scope of this paper.
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